Chapter 3 Geometry of convex functions - Meboo Publishing ...

250 CHAPTER 3. GEOMETRY OF CONVEX FUNCTIONS
α
β
α ≥ β ≥ γ
∇f(X)
Y−X
γ
{Z | f(Z) = α}
{Y | ∇f(X) T (Y − X) = 0, f(X)=α} (590)
Figure 76:(confer Figure 65) Shown is a plausible contour plot in R 2 of some
arbitrary real differentiable convex function f(Z) at selected levels α , β ,
and γ ; contours of equal level f (level sets) drawn in the function’s domain.
A convex function has convex sublevel sets L f(X) f (591). [301,4.6] The
sublevel set whose boundary is the level set at α , for instance, comprises
all the shaded regions. For any particular convex function, the family
comprising all its sublevel sets is nested. [195, p.75] Were sublevel sets not
convex, we may certainly conclude the corresponding function is neither
convex. Contour plots of real affine functions are illustrated in Figure 26
and Figure 71.